To deal with the linear coupled system (REF ),
motivated by the recent researches
[1]}, [2]} we will apply the
group Fourier transform associated with the Hermite functions (REF ). Afterwards, by constructing a
new function in the vector sense, we will
transfer the wave-heat coupled system to the first-order (in time) coupled
system. Moreover, our aim in this section is to understand asymptotic behaviors of eigenvalues and solutions in
the phase space. We will adapt the classical
phase analysis with diagonalization procedure (see, for example, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}) in the Euclidean case \(\mathbb {R}^n\)  to the Heisenberg group case \(\mathbf {H}_n\) .
